Photolithography is an essential tool in reproduction of fine patterns on a substrate, and is very widely used in production of microelectronic devices. As the design rules used in such devices become ever finer, mask designers must increasingly resort to reticle enhancement technologies, such as the use of serifs, assist lines and phase shift masks, in order to project the desired pattern onto the device substrate. The aerial image that is actually formed on the substrate is a complex function of the characteristics of the illumination source and optics that are used in the lithographic process and of diffraction and interference effects caused by the structures on the mask itself. Mask designers need simulation systems that model these effects in order to predict the pattern that will be formed on the substrate by particular arrangements of mask features.
Simulation of the aerial image is complicated by the fact that practical lithography systems use partially-coherent illumination. For optical systems that are nearly paraxial, the intensity of the aerial image at the image plane with partially-coherent illumination of the mask is given by the well-known Hopkins formula:
                                                                        I                ⁡                                  (                                      z                    ->                                    )                                            =                            ⁢                                                ∫                  Ξ                                ⁢                                                      ∫                    Ξ                                    ⁢                                                                                    g                        ⁡                                                  (                                                                                    x                              ->                                                        1                                                    )                                                                    ⁢                                                                                          ·                                                                        g                          *                                                ⁡                                                  (                                                                                    x                              ->                                                        2                                                    )                                                                    ·                      K                                        ⁢                                                                  (                                                                              z                            ->                                                    ,                                                                                    x                              ->                                                        1                                                                          )                                            ·                                                                                                                                                            ⁢                                                                                          K                      *                                        ⁡                                          (                                                                        z                          ->                                                ,                                                                              x                            ->                                                    2                                                                    )                                                        ·                                      h                    ⁡                                          (                                                                                                    x                            ->                                                    1                                                ,                                                                              x                            ->                                                    2                                                                    )                                                                      ⁢                                                      ⅆ                                                                  x                        ->                                            1                                                        ·                                                                          ⁢                                      ⅆ                                                                  x                        ->                                            2                                                                                                                              (        1        )            Here g is the transmission function of the mask; h is the mutual coherence function of the illumination source (typically the Fourier transform of the condenser aperture function); and K is the point spread function (PSF) of the projection system. The function h can be expressed in terms of a hermitian matrix. Formula (1) is a quadruple integral, taken over the two-dimensional space Ξ in the mask plane. Accurate, direct computation of this integral is computationally heavy and requires precise knowledge of the optical system properties, as expressed by K and h. In some formulations, the properties of the optical system are expressed more compactly in the form of a kernel function W=K·h.
In order to reduce the complexity of aerial image simulation, a number of authors have suggested using optimal coherent decomposition (OCD) to approximate the optical properties of the partially-coherent imaging system as an incoherent sum of a finite number of coherent imaging systems. The aerial images that would be formed by each of the coherent imaging systems are computed and summed together to give the total, simulated aerial image. Pati et al. provide a useful overview of OCD methods in “Exploiting Structure in Fast Aerial Image Computation for Integrated Circuit Patterns,” IEEE Transactions on Semiconductor Manufacturing 10:1 (February, 1997), pages 62-73, which is incorporated herein by reference. This article describes the use of “basis” (or building block) images, which correspond to certain types of integrated circuit patterns, in order to compute aerial images by OCD.
Von Bunau et al. describe a related method of OCD in “Optimal Coherent Decompositions for Radially Symmetric Optical Systems,” Journal of Vacuum Science and Technology B15:6 (November/December, 1997), pages 2412-2416, which is incorporated herein by reference. The authors show that for optical systems that are radially symmetrical, the point spread functions and pupil functions corresponding to each term in the OCD expansion are separable in polar coordinates. They derive analytical expressions for the angular dependence of these terms and an integral equation for the radial dependence.
A number of methods for aerial image simulation have been described in the patent literature. For example, U.S. Pat. No. 6,223,139, whose disclosure is incorporated herein by reference, describes a method for kernel-based fast aerial image computation for a large-scale design of integrated circuit patterns. The method is based on determining an appropriate sampling range and sampling interval for use in generating simulated aerial images of a mask pattern, so as to enhance computation speed without sacrificing accuracy. U.S. Patent Application Publication 2002/0062206, whose disclosure is also incorporated herein by reference, describes a method for fast aerial image simulation, using a kernel that is calculated based on an orthogonal pupil projection of the parameters of the optical projection system onto a basis set. A vector is calculated based on an orthogonal mask projection of the parameters of the mask onto the basis set, and the field intensity distribution in the image plane is then calculated using the kernel and the vector.